theory - Must the characteristic impedance of a transmission line be non-real for it to be lossy?

In Does conductance in the transmission line model represent a physical quantity? it came up that a non-real characteristic impedance just means the line has some loss. With real transmission lines having a little (but perhaps negligible) loss, we could expect all real transmission lines to have a characteristic impedance with at least a small imaginary component.

But is that really true? Let's say $Z = Y = 1+1j$. The characteristic impedance of this line is real:

$$\sqrt{ 1+1j \over 1+1j } = 1$$

And the attenuation constant is 1 [corrected by edit]:

$$\operatorname{Re}\sqrt{(1+1j)(1+1j)} = 1$$

Which to my understanding, means this line is lossless. But how can this be when it has both a non-zero conductance and resistance?